I agree! The footnote is intended as satire, ridiculing logicism as an account of mathematical foundations.
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Replying to @Meaningness @drossbucket and
I do think it’s inaccurate to say Presburger arithmetic is provably consistent and Peano arithmetic is not. If that’s a joke it’s likely to mislead unless explained. (Also, Presberger is decidable and complete; first-order arithmetic is recursively enumerable and complete.)
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Replying to @ESYudkowsky @Meaningness and
And whether you should *believe* the times table is about whether the times table is *sound* on a domain (which must include a correspondence theory!) and this semantic property is mathematically and practically a very different concept from consistency, a syntactic property.
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Replying to @ESYudkowsky @Meaningness and
Under other circumstances I might not nitpick that hard, but it is literally what your book is about.
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Replying to @ESYudkowsky @drossbucket and
I think you’re missing the point here, which is natural since the quote is very much out of context. (It comes about half way through the book, after a lot of conceptual machinery has been built up already.)
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Replying to @Meaningness @drossbucket and
Could be. Maybe I’m too suspicious. Out of context, it does sound a lot like some extremely common mixups about proof theory.
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Replying to @ESYudkowsky @drossbucket and
This comes after a hundred pages explaining all the ways logicism is wrong, so I think in context there will be little room for readers to misunderstand it as anything other than a satire, based on the obvious fact that there is no meaningful doubt about multiplication.
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Replying to @Meaningness @drossbucket and
I’m not worried about the reader doubting multiplication, I’m worried about spreading the common misconception that PA cannot be proven consistent. It can be, and the proof reduces to a proposition that can be made wordlessly self-evident, like the termination of a hydra game.
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Replying to @ESYudkowsky @drossbucket and
OK… this is something I am not at all worried about. (Possibly interesting generalization: the sorts of things you and I worry about are different although not disjoint.)
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Replying to @Meaningness @drossbucket and
You’ve never wandered through the dreadful wreckage of a conversation based on someone’s misunderstanding of Godel’s Theorem?
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Yes, often :( I don’t recall anyone misunderstanding the question of Peano consistency as having real-world implications, or even substantive implications for math. Do you encounter that? (I’m quite willing to believe this can happen.)
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